Circumsphere
The radius of the smallest sphere that contains all four points can be expressed as , where is not divisible by the square of any prime. Find .
The answer is 13.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
This radius is the same as the radius of the circumscribed sphere of the tetrahedron with vertices at the given points. Then, we realize that the circumcenter of this tetrahedron must lie on the lines perpendicular to the faces at the circumcenters of those faces. Because some of the faces are right triangles, it is simple to find that the circumcenter is (4,5,7). Then, we find the radius by the Pythagorean Theorem. We thus get that r = 3 1 0 , and 3+10 = 13.